Constant Speed of Light

a car moving in a straight line at constant velocity, say 30mph, turns on its headlights and yet when the speed of the photons emitted is measured by an observer standing either in front of or along side the moving vehicle, the photon measurement is still only c and not c + v ; why is that? Could you show me the mathematics to justify this?
- william (age 21)
san diego, ca. usa

First, let me try to be explicit about why this surprises you. You think instinctively that the distance that the light travels (say from one car to the rear of the next) is an objective feature whose value is the same as seen by someone in one of the cars moving at speed v or someone on the ground. You also instinctively think that the time (t) it takes for the light to go from one car to the next is the same according to any observer. If that were true, then if we saw light traveling at speed c, the speed of the light measured by someone on the cars would just be (ct-vt)/t =c-v, since from the cars' point of view the distance between them never changed during the light's trip, while we believed that the lead car went distance vt. It turns out that both these gut feelings are false.

The actual rules for seeing how the coordinates of one observer map to those of a different one are called the Lorentz transforms. () Using these rules to figure out the distance between the cars as seen in their own frame (c-v)t/sqrt(1-v²/c²)) rather than (c-v)t. The time to travel from one car to the next is t*sqrt((c-v)/c+v)) rather than t. That distance over that time still comes out c.

Mike W.

(published on 07/24/2012)